A** field** is a set F together with two binary operations, called addition and multiplication,
which satisfy the following axioms:

- 1. F under addition is an abelian group.
- This means that
- a) a + (b + c) = (a + b) + c a, b, cF (
*The Associative Law for Addition*) - b) a + b = b + a a,bF (
*The Commutative Law for Addition*) - c) an element, called 0, which satisfies a + 0 = 0 + a = a aF (
*Additive Identity*) - d) For every element aF, there exists an element denoted -a which satisfies a + (-a) = (-a) + a = 0. (
*Additive Inverse*)

- a) a + (b + c) = (a + b) + c a, b, cF (
- 2. F - {0} under multiplication is an abelian group.
- This means that
- a) a(bc) = (ab)c a, b, cF-{0} (
*The Associative Law for Multiplication*) - b) ab = ba a,bF-{0} (
*The Commutative Law for Multiplication*) - c) an element, called 1, which satisfies a1 =1a = a aF - {0} (
*Multiplicative Identity*) - d) For every element aF-{0}, there exists an element denoted a
^{-1}which satisfies a(a^{-1}) = (a^{-1})a = 1. (*Multiplicative Inverse*)

- a) a(bc) = (ab)c a, b, cF-{0} (
- 3. a, b, cF we have a(b + c) = ab + ac (
*The distributive law of multiplication over addition*). - 4. 0a = a0 = 0 aF.

- 1. x < x is never true. (
*irreflexivity*). - 2. If x < y and y < z then x < z (
*transitivity*) - 3. Either x < y, x = y or y < x (
*trichotomy*) - 4. If x < y, then x + z < y + z.
- 5. If x < y and 0 < z, then xz < yz.

Let A be a subset of an ordered field F. We say that sF is a **least upper bound** or
**supremum** of A in F iff

- 1. s is an upper bound for A.
- 2. st for all upper bounds t of A.

- 3. i is a lower bound for A.
- 4. ri for all lower bounds r of A.

An ordered field F is** complete** iff every nonempty subset of F that has an upper bound in F
has a supremum that is in F.

The rationals are not complete, but the reals form a complete ordered field. In fact, it can be shown that any complete ordered field is just a copy (with the elements renamed) of the reals, so in this sense there is only one complete ordered field.

We will now give the definitions that permit us to state three important theorems about the reals.

For any real number a, if is a positive real number, the **-neighborhood of a** is the
set N(a, ) = {x : | x - a | < } = (a -, a +).

For a set A, a point x is an** interior point of A** iff there exists a > 0 such
that N(a, )A. The set A is **open** iff every point of A is an interior point of A.
The set A is **closed** iff its complement is open.

Let B be a set. A collection of setsis a** cover** of B iff B is contained in the union of
all the sets of. A **subcover** offor B is a subcollection of the sets ofwhich also cover B.

A set A is **compact** iff every cover of A by open sets has a finite subcover.

An element x is an **accumulation point of the set A** iff for all > 0, N(x,) contains
a point of A distinct from x.

The set of accumulation points of a set A is called the **derived set** of A and denoted A'.

A **sequence** is a function whose domain is. We shall be talking only about real
sequences, which are sequences whose codomain is.

A **sequence is bounded** iff its range is a bounded subset.

A sequence {x_{n}} has a **limit L**, or {x_{n}} **converges to L**, iff for every real> 0, there
exists a natural number N such that if n > N, then | x_{n} - L | <. If no such L exists,
the sequence **diverges**.

A sequence {x_{n}} is **increasing** iff for all n, m, x_{n}x_{m} whenever n < m. A
**decreasing** sequence requires x_{n} x_{m} whenever n < m. A **monotone** sequence is one that is either increasing or decreasing.

Now we are ready to state the three theorems. One can use the completeness of to prove the famous:

**Heine-Borel Theorem**: A subset A ofis compact iff A is closed and bounded.

This theorem can then be used to prove the:

**Bolzano-Weierstrass Theorem**: Every bounded infinite subset ofhas an accumulation point in.

In turn this theorem is used to prove the:

**Bounded Monotone Sequence Theorem**: Every bounded monotone sequence has a limit L in.

Finally, it can be shown that if an ordered field satisfies the Bounded Monotone Sequence Theorem then the field is complete.

The circular series of proofs of these theorems shows that all of them are equivalent to the completeness of.